Need help adjusting shipping costs to per product.
Products that share the same weight do not always have the same shape or fit in the same boxes.
Magento is not making this distinction. How can I adjust it so we can reflect the accurate price for shipping based on the box that's being used?
Magento will not do this out the box. You will need to look for a custom extension that does it.
Have a look through the shipping section on magento connect and see if any suit your needs.
The only shipping extensions I have used are WebShopApps - Product Matrix and WebShopApps - Premium Matrix. As I mentioned there are plenty others out there.
They (the 2 mentioned above) are mainly CSV based and require some patience to get the right rates appearing - they have an extensive wiki detailing how to fill them out. As for USPS - since I'm UK based I've never had any need to integrate with them so can't recommend anything based on them.
Magento connect does have a Star and a Heat System so you can see which plugins are popular and which have good reviews - many of them also offer demo stores to test things out on too.
You can also have a look at this extension that does what you need - Shipping per product